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Theorem uspgrushgr 26541
Description: A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)
Assertion
Ref Expression
uspgrushgr (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)

Proof of Theorem uspgrushgr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2778 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2778 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2isuspgr 26518 . . . 4 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4 ssrab2 3908 . . . . 5 {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5 f1ss 6358 . . . . 5 (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
64, 5mpan2 681 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
73, 6syl6bi 245 . . 3 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})))
81, 2isushgr 26426 . . 3 (𝐺 ∈ USPGraph → (𝐺 ∈ USHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})))
97, 8sylibrd 251 . 2 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph))
109pm2.43i 52 1 (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  {crab 3094  cdif 3789  wss 3792  c0 4141  𝒫 cpw 4379  {csn 4398   class class class wbr 4888  dom cdm 5357  1-1wf1 6134  cfv 6137  cle 10414  2c2 11435  chash 13441  Vtxcvtx 26361  iEdgciedg 26362  USHGraphcushgr 26422  USPGraphcuspgr 26514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5027
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fv 6145  df-ushgr 26424  df-uspgr 26516
This theorem is referenced by:  uspgrupgrushgr  26543  usgredgedg  26593  vtxdusgrfvedg  26856  1loopgrvd2  26868  isomuspgr  42761
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